An efficient method for counting short cycles in the Tanner graphs of quasi cyclic (QC) protograph low-density parity-check (LDPC) codes is presented. The method is based on the relationship between the number of short cycles in the graph and the eigenvalues of the directed edge matrix of the graph. We demonstrate that for a QC protograph LDPC code, the complexity of computing such eigenvalues can be reduced significantly by representing the directed edge matrix as a block circulant matrix. Numerical results are presented to show the lower complexity of the proposed method compared to the existing algorithms for counting short cycles. These results also reveal that QC LDPC codes on average have a superior short cycle and girth distribution compared to similar randomly constructed codes.

Counting short cycles, Girth, Girth distribution, Low-density parity-check (LDPC) codes, Protograph LDPC codes, Quasi cyclic (QC) LDPC codes, Short cycle distribution, Short cycles
IEEE Communications Letters
Department of Systems and Computer Engineering

Karimi, M. (Mehdi), & Banihashemi, A. (2012). Counting short cycles of quasi cyclic protograph LDPC codes. IEEE Communications Letters, 16(3), 400–403. doi:10.1109/LCOMM.2012.020212.112311